Question

For the function f whose graph is given, state the value of the given quantity if it exists. If it does not exist, explain why.

$\left(a\right)\underset{x\to 0}{\lim }f\left(x\right)$

$\left(b\right)\underset{x\to 3^{-}}{\lim }f\left(x\right)$

$\left(c\right)\underset{x\to 3^{+}}{\lim }f\left(x\right)$

$\left(d\right)\underset{x\to 3}{\lim }f\left(x\right)$

$\left(e\right)f\left(3\right)$

Short answer

(a)

The value of limit of the given function is$3.$

Step-by-step solution

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches $0$is $3$

$\underset{x\to 0}{\lim }f\left(x\right)=3$

Therefore, our final answer is $3.$

(b)

The value of limit of the given function is$3.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches $3^{-}$ from the left side is$3.$

$\underset{x\to 3^{-}}{\lim }f\left(x\right)=3$

Therefore, our final answer is $3.$

(c)

The value of limit of the given function is$3.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Since the left- and right-hand limits are different, we conclude that

$\underset{x\to 3^{+}}{\lim }f\left(x\right)=2$

Therefore, our final answer is $2.$

(d)

The value of limit of the given function is$3.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Since the left- and right-hand limits are the same, so we have

$\underset{x\to 3}{\lim }f\left(x\right)=3$

Therefore, our final answer is $3.$

(e)

The value of limit of the given function is$\text{0}.$

Step 1. Given information.

The graph of the limits are given here,

Step 2. Formula used.

Here, we use the concept of limits from a graph.

Step 3. Simplification.

Limit of f as x approaches $3$ is $0$

$f\left(3\right)=0$

Therefore, our final answer is$0.$